Portfolio optimisation in the Indian stock market - industry sector analysis

Gupta, R.1

Parikshit K. Basu2

Abstract

Diversification always reduces non-systematic risk within a portfolio to a certain

extent. At the same time, selection of individual items or industry sector influences

returns. Optimum portfolio selection within a capital market is primarily based on the

best risk-return trade-off among the industry sectors. Literature suggests that much of

market volatility can be attributed to substantial increase in sector specific and sub-

sector specific risks. Performance of the economy influences industry sector returns

differently and changes over time periods. Thus, changing pattern of correlations

between sectors is vital for portfolio optimization purpose. The present study has

estimated the dynamics of correlations of stock market returns between industry

sectors in India using Asymmetric DCC GARCH model and tested efficient portfolios

that generates returns above the market average. Use of Asymmetric DCC GARCH

model helps in capturing the dynamics of correlations and as such a better estimate of

the expected future correlations resulting in a portfolio that reflects future outcome

more closely. If the risk adjusted returns of the industry selected portfolio is more

than the index returns, we could argue that there is value in industry selection and

accurate measurement of the correlations help generate these excess returns. Using

daily market data for the period April 1997 to April 2007 on a sample of 10 industry

sectors selected randomly indicates that investors can substantially improve their

reward to risk as compared with the market returns. Sharpe ratio of the optimised

portfolio improves to 0.994 (for optimised portfolio) from 0.527 (for S&P Nifty

index).

1 Gupta, R., School of Commerce, Faculty of Business & Informatics, Central Queensland University, Bruce Highway, Rockhhampton, QLD 4702, Australia, Tel: +61749309158, Fax: +61749309700, Email: R.gupta@cqu.edu.au2 School of Marketing & Management, Charles Sturt University, Bathurst, NSW 2795 Australia, Tel.: +612 63384577, Fax: +612 63384769, Email: pbasu@csu.edu.au

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Introduction

Background

Diversification always reduces non-systematic risk within a financial assets

portfolio to a certain extent. Any textbook dealing with the area of investments

includes basic analysis on portfolio diversification. Common objective of

financial investors is to achieve an optimal risk-return combination. It can be

achieved either by maximising return with an accepted level of risk or by

minimising risk with an acceptable rate of return. Diversification impacts risk

component of the portfolio in particular. It implies a spread of investments and

allows a middle road through the highs and lows of market performance. In

other words, diversification allows an opportunity for investments to grow with

minimum volatility. Securities behave differently from one another within the

same market based on its own performance, industry/sector conditions, national

and international factors and so on. Literature suggests that much of market

volatility can be attributed to substantial increase in sector specific and sub-

sector specific risks (Black et al 2002).

In the last three decades, a large number of countries had initiated financial

reform process to open up their economies and integrated into the global

economy. India is one of the late entrants – the reform process officially started

in 1991 only. The Indian stock market is possibly one of the oldest in Asia, but

remained at a small scale and largely outside the global integration process until

the late 1980s (Wong et al 2005). The major stock market in India located in

Mumbai (formerly known as Bombay) has always played the dominant role in

the equity market in India. It has been traditionally governed by brokers leading

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to conflict of interest situation between the interest of common investors and

those of brokers/owners of stock exchanges (Datar & Basu 2004). Reforms in

equity market in India commenced slightly earlier than the overall reforms - in

mid-1980s. With the establishment of National Stock Exchange (NSE), a new

institutional structure was introduced in India that could ensure smooth

functioning of market through a combination of new technology and efficient

market design. The Securities Exchange Board of India (SEBI) was set up as a

market regulator with statutory powers to control and supervise operations of all

participants in the capital market viz. stock exchanges, stock brokers, mutual

funds and rating agencies. The development of debt market is another

significant development, which has been facilitated by deregulation of

administered interest rates. Opening of stock exchange trading to Foreign

Institutional Investors (FIIs) and permission of raising funds from international

market through equity linked instruments have introduced a degree of

competition to domestic exchanges and other market participants. Coverage of

stock exchanges in India as primary and secondary markets is expanding

steadily. NSE and BSE ranked 3rd and 6th among the stock exchanges in the

world in 2006 in terms of number of transactions (GOI 2007).

Interestingly, stability in prices for the Bombay Stock Exchange (BSE) in India

was considered to be an important feature. During the period 1987 to 1994,

average annual price fluctuations of ordinary shares on BSE were 25.1% as

compared with London Stock Exchange (22%), and the New York Stock

Exchange (23.9%) (Poshakwale 1996). Regarding efficiency and level of global

integration of the Indian stock market, findings are diversified and often

contradictory. An early empirical study based on 11 years data 1963-73 even

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found evidence of random walk in BSE stocks that could be compared with

behaviour of stock prices in the markets like LSE and NYSE (Sharma &

Kennedy 1977). Wong et al (2004) observed that the Indian stock market is

integrated with mature markets and sensitive to their dynamics in the long run.

In the short-run, the US and Japan markets influenced the Indian market but not

vice versa. In particular, the Indian stock markets appear to have integrated

more with the global markets since late 1990s due to lower barriers on foreign

investments and growing presence of Indian firms in the international markets

(Banerjee & Sankar 2006). However, more recent data covering the period

1991-2006 for two major equity markets in India (BSE and NSE) found no

evidence of random walk model (Gupta & Basu 2007). Thus, evidences are

contradictory. In any case, with or without explicit validity of random walk

model, benefits of diversification cannot be denied. There is no study on

benefits of diversification of the Indian stock market.

Performance of the economy influences industry sector returns differently and

changes over time periods. Studies found strong evidence of relationships

between performances of the Indian Stock Market and macroeconomic

variables such as inflation, domestic output growth and macroeconomic

management (Naka et al 1999). After experiencing very moderate growth till

1980s India largely liberalised its economy in the early 1990s. The reforms

undertaken by the Indian government in 1991 have resulted in an increase in the

economic growth. In the last three years (2004-05 to 2006-07), average annual

real GDP growth rate in India was about 8.4%. However, sectoral growth rates

varied significantly during the period. Services contributed more than two-

thirds of this GDP growth and the entire residual contribution came from

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industry. As a result, in 2006-07, the share of agriculture in GDP declined to

18.5% and the share of industry and services improved to 26.4% (GOI 2007).

Within the services sector, IT and IT-enabled activities, trade, hotels, transport

and communication services performed much better than the others. Within the

industry sector, manufacturing, textile products, chemicals, non-electrical

equipments etc. performed better than the average in recent years (GOI 2007).

Accordingly, stocks from different sectors and sub-sectors also performed

differently during the period and need to be viewed with proper care. Each

sector is unique in its own way and so are the companies operating in each

sector. All securities behave differently from one another – even within a sector.

Still there exist some commonalities within each sector. Because investments in

each sector react differently to market conditions and other factors, in general it

may be worthwhile to maintain a sector diversified portfolio in order to balance

out the ups and downs and to optimise the portfolio. Again, to obtain

diversification benefits correlation between the two sets assets must be less than

perfect and should be considered in a dynamic setting. Thus, changing pattern

of correlations between sectors is vital for portfolio optimization purpose.

Research objectives

The present study has made an attempt to estimate the dynamics of correlations

of stock market returns between industry sectors in India using Asymmetric

DCC GARCH model and tested efficient portfolios that generates returns above

the market average. Use of Asymmetric DCC GARCH model helps in capturing

the dynamics of correlations and as such a better estimate of the expected future

correlations resulting in a portfolio that reflects future outcome more closely. If

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the risk adjusted returns of the industry selected portfolio is more than the index

returns, we could argue that there is value in industry selection and accurate

measurement of the correlations help generate these excess returns.

Research Methodology

DCC Model

In this study we estimate the time varying correlations using the Asymmetric

Dynamic Conditional Correlation (DCC) model of Cappielo, Engle and

Sheppard (2006). This model is an introduction of asymmetric term in original

DCC model of Engle (2002) as modified by Sheppard (2002) as a general

model. The conditional correlation between two random variables r1 and r2 that

have mean zero can be written as:

(1)

Substituting the above into equation (1) we get:

(2)

Using GARCH(1,1) specification, the covariance between the random variables

can be written as:

(3)

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The unconditional expectation of the cross product is , while for the

variances

= 1

The correlation estimator is:

(4)

This model is mean reverting if α + β < 1. The matrix version of this model is

written as:

111 )()1( tttt QSQ (5)

Where S is the unconditional correlation matrix of the disturbance terms and

. The log likelihood for this estimator can be written as:

(6)

Where and Rt is the time varying correlation matrix.

As this model does not allow for asymmetries and asset specific news impact

parameter, the modified model Cappiello, Engle and Sheppard (2006) for

incorporating the asymmetrical effect and asset specific news impact is:

(7)

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Where A, B and G are diagonal parameter matrixes, nt = I[εt < 0]o εt (with o

indicating Hadamard product), . For and , expectations are

infeasible and are replaced with sample analogues, and

, respectively. is a diagonal matrix with the

square root of the ith diagonal element of Qt on its ith diagonal position. In this

paper we only look for the asymmetrical effects and not the asset specific news

impacts.

Efficient Portfolios

The efficient frontier is defined as the set of portfolios that exhibit the minimum

amount of risk for a given level of return or the highest return for a given level

of risk, and lies above the global minimum variance portfolio. Elton, Gruber

and Padberg (1976) show one is able to use a simple decision criterion to reach

a optimal solution to the portfolio problem by assuming that a risk-free asset

exists, and either the single index model adequately describes the variance-

covariance structure, or that a good estimate of pair wise correlations is a single

figure. This simple criterion not only allows one to determine which stocks to

include but how much to invest in each.

The first approach utilises the single-index model to construct optimal

portfolios. Where returns are determined as follows:

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Assuming that short selling is possible, the task would be to find the

unconstrained vector of relative weights for each security so that the Sharpe

ratio is maximised. That is:

(8)

Given that

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These equations are substituted into the Sharpe ratio equation and in order to

maximise the Sharpe ratio it is necessary to take the derivative of the Sharpe

ratio with respect to each and set it equal to zero. The derivation yields the

amount of the portfolio that should be invested in any security as:

Thus by applying the above equation then one is able to determine the

respective weightings for each security within the portfolio and to find the

optimal portfolio's risk and return measures. That is, the risk and returns are

obtained by substituting the respective weights found for each security into the

returns and variance formula given in (9) and (10) respectively3.

Data requirement

For this study we use monthly returns of the National Stock Exchange (NSE)

and the monthly returns of the different sector indices for the period April 1997

to April 2007. In order to calculate the volatility of the respective index, we use

3 The model is for no restrictions on short sale, the standard optimization problem can be written as:

subject to the constraint : ; if the short selling is not allowed

the additional constraint will be of non-negative weights, expressed as 0 Xi 1. Similar minimum or maximum weight restrictions are imposed when introducing restrictions of minimum investments into Australian index and or maximum restrictions into emerging market indexes.

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daily prices to calculate the daily returns and the daily average volatility of each

market index returns. We calculate monthly volatility (Volatilitym = Daily

volatility X √n, where m represents period and n number of trading days in the

period) of each market on the basis of actual number of trading days in the

month for the emerging market. We use DataStream for index values of the

respective equity indexes. Indexes included in the study are; Abrasives, Air

conditioners, Automobiles 2 wheeler, Aluminium, Construction, Durables,

Refinery, Software, Synthetic Textiles and Trading.

Table 1 lists sector returns and their summary statistics for the 10 market

sectors. Mean daily returns of the market sectors included in this study varies

from 0.5% for Durables to 1.6% for Software sector. There are major

differences in the minimum and maximum daily returns for the sectors and as

such major differences in the variances.

Table 1 : Summary Statistics

Abrasives

Air conditioners

Aluminium

Automobiles

Construction

Durables

Refinery

Software

Synthetic Textiles

Trading

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Analysis of results

The unconditional correlations between sectors for the sample period are

presented in Table 2. Lowest correlations were observed between Abrasives and

Software, 0.167 for the sample period and highest between Aircondioners and

Durables, 0.472.

  Abrasives Aircon Alum Autos2 Construc Durables Refinery Softwa SyntTex TradingAbrasives 1.000Aircon 0.219 1.000Alum 0.205 0.331 1.000Autos2 0.189 0.339 0.392 1.000Construc 0.264 0.354 0.376 0.378 1.000Durables 0.232 0.472 0.399 0.414 0.426 1.000Refinery 0.232 0.377 0.437 0.386 0.441 0.443 1.000Softwa 0.167 0.357 0.299 0.329 0.360 0.446 0.323 1.000SyntTex 0.251 0.442 0.426 0.398 0.436 0.532 0.429 0.357 1.000Trading 0.224 0.309 0.291 0.285 0.363 0.378 0.353 0.332 0.357 1.000

Table 2 : Unconditional Correlations

The focus of this paper is to use conditional correlation estimates (as against

unconditional ones) as being more accurate. These correlations are estimated

using Asymmetric Dynamic Conditional Correlation Model (ADCC GARCH)

as discussed earlier in the methodology section. Table 3 presents the conditional

correlations calculated using ADCC GARCH model. Lowest correlations of

0.1221 are between Abrasives and Aluminium and highest of 0.5747 between

Construction and Refinery. Direct comparison of unconditional correlations

from table 2 and conditional correlations in table 3 is not meaningful as the

argument in favour of conditional correlations lies in theoretical strengths of

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Asymmetric DCC GARCH model which is expected to produce better estimates

for correlations as the model allows for the correlations to change over time.

  Abrasives Aircon Alum Autos2 Construc Durables Refinery Softwa SyntTex TradingAbrasives 1

Aircon 0.2213 1

Alum 0.1221 0.2851 1

Autos2 0.2097 0.3823 0.2680 1

Construc 0.3550 0.3328 0.4662 0.4763 1

Durables 0.2181 0.3583 0.3455 0.4321 0.4460 1

Refinery 0.2352 0.4069 0.4081 0.4394 0.5747 0.4542 1

Softwa 0.2042 0.3756 0.3597 0.5338 0.4380 0.5653 0.5116 1

SyntTex 0.2330 0.4169 0.3054 0.4537 0.4025 0.3419 0.4169 0.3332 1

Trading 0.2369 0.3733 0.3239 0.4148 0.4782 0.4520 0.3764 0.3671 0.3479 1

Table 3 : Conditional Correlations (ADCC GARCH)

In the next step we look at the mean returns and the Sharpe ratios for the NSE

index. These have been used to compare the portfolio returns to assess if the

portfolio constructed using the model performs better than the broad based index.

Table 4 includes the portfolio mean returns for National Stock Exchange (NSE) of

India. We will compare the returns of the optimal portfolio using sector indexes for

the market and conditional correlations estimated using ADCC GARCH model.

India only portfolio has a mean return of 18.91% with a standard deviation of

24.48% and Sharpe ratio of 0.52 based on a risk free rate of 6% and a 50%

probability of achieving the target mean returns.

Table 4: S & P Nifty Index

Risk free return 6%

Mean annual returns 18.91%

Standard deviation 24.48%

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Sharpe ratio 0.52

Probability of achieving mean returns

50.00%

The main focus of the paper is the use of computationally efficient model4 for

estimating the correlations. As such we use conditional correlations in

optimisation model to construct optimised portfolio and compare the returns of

this portfolio with the S & P Nifty index. Proportion invested into each market

sector, annualised returns and standard deviation of the portfolio are presented

in Table 6 and the estimated Sharpe ratios are presented in Table 7.

Table 6

Indian portfolio of different market sectors using ADCC GARCH correlations

Weight Returns % Standard Deviation %

Abrasives 12.89 30.59 38.54

Airconditioning 8.05 34.44 36.93

Aluminium -9.29 13.60 36.25

Automobiles(2) -3.21 20.19 29.72

Construction 6.60 33.81 41.82

Durables -6.79 13.34 41.47

Refinery -9.94 16.91 36.22

Software 13.40 43.52 44.99

Synthetic Textiles -4.45 17.98 33.90

Trading -7.27 28.13 37.68

4 Results for portfolios with unconditional correlations are not presented here but can be reqested from the authors.

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PORTFOLIO 32.69 26.58

Table 7 : Comparison of Sharpe Ratios

Portfolio Mean Return Sharpe Ratio* ProbabilityS & P Nifty 18.91% 0.527 50.00%Optimal portfolio

32.69% 0.994 69.545%

* Proxy for Risk free rate is average money market rates for April 2007, acquired from Reserve bank of India (6.0%).

Comparisons of Sharpe ratio (Table 7) indicate that the optimised portfolio

using ADCC GARCH correlations is a superior portfolio as compared with the

S & P Nifty index6. The mean return of the optimised portfolio is 32.69% which

is more than double than that of NSE. Standard deviations of these portfolios

are very similar, 24.48% for NSE and 37.68%. Results clearly indicate the

expected returns of the optimised portfolio can be improved without

significantly compromising on the risk. Using Sharpe ratio for objectively

comparing the returns we see that Sharpe ratio improves from 0.527 to 0.994.

This is a significant improvement from the NSE returns.

Conclusions

The importance of industry selection has been tested in this study using a

sample of 10 industry sectors for equity markets in India. It can further be

argued that if industry selection is important so the correlations within the

industry pairs are important for portfolio optimization purpose so as to enhance

5 This is the probability of achieving the target return of 18.91% in both cases.6 S & P Nifty is a capitalisation weighted index of 50 stocks representing 21 industry sectors from the National Stock Exchange, India

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portfolio returns. If correlations within the industry are important, than an

accurate estimate of correlations becomes evident and Theoretically

Asymmetric DCC GARCH estimates should provide us with a better estimate

of correlations and the results indicate correlations do change over time. On

theoretical grounds this study recommends using Asymmetric DCC GARCH

model for estimates of correlations. By using Asymmetric DCC GARCH model

for estimating correlations and using these correlations in portfolio optimization

process we can enhance portfolio returns of the domestically diversified

portfolio.

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